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Half-Plane of Absolute Convergence

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Half-Plane of Absolute Convergence 221- Extremal values of Dirichlet L-functions in the half-plane of absolute convergence par JÖRN STEUDING RÉSUMÉ. On démontre que, pour tout 03B8 réel, il existe une infinité de s = 03C3 + it avec 03C3 ~ 1 + et t ~ +~ tel que log log t log log log log t Re {exp i 03B8) log L (s, } ~ log La démonstration est basée sur une version effective du théorème de Kronecker sur les approximations diophantiennes. ABSTRACT. We prove that for any real 03B8 there are infinitely many values of s = 03C3 + it with 03C3 ~ 1 + and t ~ +~ such that Re { exp ( i 03B8) log L (s, } ~ log The proof relies on an effective version of Kronecker’s approxima- tion theorem. 1. Extremal values Extremal values of the Riemann zeta-function in the half-plane of abso- lute convergence were first studied by H. Bohr and Landau [1]. Their results rely essentially on the diophantine approximation theorems of Dirichlet and Kronecker. Whereas everything easily extends to Dirichlet series with real coefficients of one sign (see [7], §9.32) the question of general Dirichlet se- ries is more delicate. In this paper we shall establish quantitative results for Dirichlet L-functions. Let q be a positive integer and let X be a Dirichlet character mod q. As usual, denote by s = Q + it with a, t E IEB, i2 = -1, a complex variable. Then the Dirichlet L-function associated to the character X is given by - . where the product is taken over all primes p; the Dirichlet series, and so the Euler product, converge absolutely in the half-plane 7 > 1. Denote by Manuscrit regu le 9 juillet 2002. 222 xo the principal character mod q, i.e., Xo(n) = 1 for all n coprime with q. Then Thus we may interpret the well-known Riemann zeta-function ((s) as the Dirichlet L-function to the principal character Xo mod 1. Furthermore, it follows that L(s, Xo) has a simple pole at s = 1 with residue 1. On the other side, any L(s, x) xo is regular at s = 1 with L(1, x) # 0 (by Dirichlet’s analytic class number-formula). Since L(s, X) is non-vanishing in Q > 1, we may define the logarithm (by choosing any one of the values of the logarithm). It is easily shown that for Q > 1 Obviously, IlogL(s,x)l::; L(a,xo) for a > l. However Theorem 1.1. For any E > 0 and any real 0 there exists a sequence of s = a + it with a > 1 and t -> +oo such that In particular, In spite of the non-vanishing of L(s, X) the absolute value takes arbitrarily small values in the half-plane Q > 1! The proof follows the ideas of H. Bohr and Landau [1] (resp. [8], §8.6) with which they obtained similar results for the Riemann zeta-function (answering a question of Hilbert). However, they argued with Dirichlet’s homogeneous approximation theorem for growth estimates of 1((8)1] and with Kronecker’s inhomogeneous approximation theorem for its reciprocal. We will unify both approaches. Proof. Using (2) we have for x > 2 223 Denote by cp(q) the number of prime residue classes mod q. Since the val- ues x(p) are cp(q)-th roots of unity if p does not divide q, and equal to zero otherwise, there exist integers Ap (uniquely determined mod p(q)) with Hence, I I ’.A/ I In view of the unique prime factorization of the integers the logarithms of the prime numbers are linearly independent. Thus, Kronecker’s approxi- mation theorem (see [8], §8.3, resp. Theorem 3.2 below) implies that for any given integer w and any x there exist a real number T > 0 and integers hp such that Obviously, with w - oo we get infinitely many T with this property. It follows that . , ,,¿- , provided that w > 4. Therefore, we deduce from (3) / I I I resp. in view of (2). Obviously, the appearing series converges. Thus, sending w and x to infinity gives the inequality of Theorem 1.1. By (1) we have for a -t 1+. Therefore, with 0 = 0, resp. 0 = 7r, and a - 1+ the further assertions of the theorem follow. D The same method applies to other Dirichlet series as well. For example, one can show that the Lerch zeta-function is unbounded in the half-plane 224 of absolute convergence: if a > 0 is transcendental; note that in the case of transcendental a the Lerch zeta-function has zeros in Q > 1 (see [3] and [4]). In view of Theorem 1.1 we have to ask for quantitative estimates. Let 7r(x) count the prime numbers p x. By partial summation, The prime number theorem implies for x > 2 By the second mean-value theorem, for some ~ E (x, y) . Thus, substituting ~ by x and sending y - oo, we obtain the estimate ~ l-n oo. This gives in (6) Substituting (7) in formula (8) yields Re (exp(10) log L (a + iT, x)~ Let then x tends to infinity as a - 1-~-. We obtain for x sufficiently large The question is how the quantities w, x and T depend on each other. 225 2. Effective approximation ° H. Bohr and Landau [2] (resp. [8], §8.8) proved the existence of a T with 0 T exp(N6) such that where p, denotes the v-th prime number. This can be seen as a first effective version of Kronecker’s approximation theorem, with a bound for T (similar to the one in Dirichlet’s approximation theorem). In view of (5) this yields, in addition with the easier case of bounding ,(( s) from below, the existence of infinite sequences S:i: = a± + it± with cr~ 2013~1+ +oo for which where A > 0 is an absolute constant. However, for Dirichlet L-functions we need a more general effective version of Kronecker’s approximation theorem. Using the idea of Bohr and Landau in addition with Baker’s estimate for linear forms, Rieger [6] proved the remarkable Theorem 2.1. Let v, N E N, b E ~,1 w, U E R. Let pi ... PN be prirrae numbers (not necessarily consecutive) and Then there exist hv E Z, 0 v N, and an effectively computable number C = C (N, PN) > 0, depending on N and pN only, with We need C explicitly. Therefore we shall give a sketch of Rieger’s proof and add in the crucial step a result on an explicit lower bound for linear forms in logarithms due to Waldschmidt ~9~. Let K be a number field of degree D over Q and denote by the set of logarithms of the elements of {0}, i.e., If a is an algebraic number with minimal polynomial P(X) over Z, then define the absolute logarithmic height of a by I -l1 note that h(a) = loga for integers a > 2. Waldschmidt proved 226 Theorem 2.2. E LK and /3v for v = 1, ... , N, not all equal zero. Define av = for v = 1, ... , N and Let E, W and v N, be positive real numbers, satisfying and Finally,, define Vv = 1} for v = N and v = N - 1, with Y+ = 1 in the case N = l. If A ~ 0, then with This leads to Theorem 2.3. With the notation of Theorem 2.1 and under its assump- tions there exists an integer ho such that (11) holds and if PN is the N-th prime number, then, for any E > 0 and N sufficiently large, Proof. For t define 227 With > we have By the multinomial theorem, Hence, for 0 B 6 R and kEN By the theorem of Lindemann AT vanishes if and only if jv = jv for v = -1, 0, ... , N. Thus, integration gives , and if jv for some v E {20131,0,..., N~ . In the latter case there exists by Baker’s estimate for linear forms an effectively computable constant A such that _ 228 Setting have, with the notation of Theorem 2.2, = = = = for v N. We may take E = 1, W logpjv, Vi 1 and Vv log pv 2, ... , If N > 2, Theorem 2.2 gives ...T .... I Thus we may take Hence, we obtain Since application of the Cauchy Schwarz-inequality to the first multiple sum and of the multinomial theorem to the second multiple sum on the right hand side of (16) yields I B 2 Setting A defined by we obtain 229 This gives note that By definition 7BJ Therefore, using the triangle inequality, and arbitrary t E 118. Thus, in view of (17) If hv denotes the nearest integer to then For v = 0 this implies 17 - ho~ G ~. Replacing T by ho yields Putting we get Substituting (15) and replacing b-1 by b, the assertion of Theorem 2.1 fol- lows with the estimate (12) of Theorem 2.3; (13) can be proved by standard estimates. D 3. Quantitative results We continue with inequality (9). Let pnr be the N-th prime. Then, using Theorem 2.3 with N = 7r(x), v = u" = 1, and yields the existence of T = 27rho with such that (4) holds, as N and x tend to infinity. We choose cv = log log ~, then the prime number theorem and (18) imply log x = log N + 0 (log log N), log N > log log log T + 0 (log log log log T) . Substituting this in (9) we obtain 230 Theorem 3.1.
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